Problem: Integrate. $ \int 4\csc(x)\cot(x)\,dx $ Choose 1 answer: Choose 1 answer: (Choice A) A $4\csc x + C$ (Choice B) B $4\cot x + C$ (Choice C) C $-4\csc x + C$ (Choice D) D $-4\cot x + C$
We need a function whose derivative is $4\csc(x)\cot(x)$. We know that the derivative of $\csc(x)$ is $-\csc(x)\cot(x)$, so let's start there: $\dfrac{d}{dx} \csc(x) = -\csc(x)\cot(x)$ Now let's multiply by $-4$ : $\dfrac{d}{dx}\left[ -4\csc(x)\right] = -4\dfrac{d}{dx}\csc(x) =4\csc(x)\cot(x)$ Because finding the integral is the opposite of taking the derivative, this means that: $ \int 4\csc(x)\cot(x)\,dx =-4 \csc(x)\, + C$ The answer: $-4 \csc(x)\, + C$